As mentioned above, the upper bound (REF ) can also be readily obtained via the classical localized Rademacher complexity arguments [1]}, [2]} or the more recently introduced offset Rademacher complexity [3]}. Crucially, the lower bound (REF ) and the upper bound (REF ) differ by a factor of \(d\)  in the worst case. It appears that the suboptimal dependence on the boundedness constants \(br\)  and \(m\)  arises due to an application of the Ledoux-Talagrand contraction inequality [4]}. In particular, when \({Y}_{L_{\infty }(P_{r})} \le m\) , the quadratic loss is \(2(br + m)\) -Lipschitz on \(\mathcal {W}_{b}\) , and the constant \((br + m)^{2}\)  propagates into the resulting upper bounds. It is well-known that in the context of unbounded distributions, the application of the contraction argument can yield suboptimal bounds [5]}; our work shows that the same is true in the classical bounded setup. In order to avoid the contraction step, we base our analysis on two components. First, we reduce our proof to the analysis of the localized multiplier and quadratic processes as in [6]}, [5]}. Second, we use a version of Rudelson's inequality [8]} for sums of rank-one operators due to Oliveira [9]} to analyze the localized quadratic process. We believe that our approach can also be used in the case of unbounded distributions: there is a version of Oliveira's bound for unbounded matrices [10]} that can be used to replace the assumption \(\Vert X\Vert  \le r\)  with a strictly weaker sub-Gaussian tail assumption on the norms \(\Vert X\Vert \) ; this could be seen as a step towards incorporating unbounded distributions within our framework while not relying on moment equivalence assumptions discussed above.
